The Complexity of Sampling Histories

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The Complexity of Sampling Histories

• Scott Aaronson, UC Berkeley
• http//www.cs.berkeley.edu/aaronson
• August 5, 2003

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Words You Should Stop Me If I Use
Words Ill Stop You If You Use
Words To Be Careful With

• polysize
• oracle
• relativizing
• zero-knowledge
• P-complete
• nonuniform

holonomy gauge SU(2) intertwinor kinematical Lagra
ngian
loop string
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Outline

• Why you should stay up at night worrying about
  quantum mechanics
• Dynamical quantum theories
• Solving Graph Isomorphism by sampling histories
• Search in N1/3 queries (but not fewer)

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What we experience
Quantum mechanics
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Assumption
Time
Quantum state of the universe
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A Puzzle

• Let OR? you seeing a red dot
• OB? you seeing a blue dot

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The Goal
Quantum state
Quantum state
Unitary matrix
Probability distribution
Probability distribution
Stochastic matrix
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Why Look for This?

• Quantum theory says nothing about multiple-time
  or transition probabilities

• Reply
• But we have no direct knowledge of the past
  anyway, just records

• Then what is a prediction, or the output of a
  computation, or the utility of a decision?

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Bohms Theory

• Gives a deterministic evolution rule for
  particle positions and momenta

• But doesnt make sense for discrete observables

• Mathematicianly approach Study the set of all
  discrete dynamical rules, without presupposing
  one of them is true

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Our Results

• We define dynamical theories for obtaining
  classical histories, and investigate what axioms
  they can satisfy

• We give evidence that by examining a history,
  one could solve problems that are intractable
  even for a quantum computer
• Graph Isomorphism and Approximate Shortest
  Lattice Vector in polynomial time
• Unordered search in N1/3 steps instead of N1/2

• We obtain the first model of computation
  slightly more powerful than quantum computing

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Dynamical Theory

• Fix an N-dimensional Hilbert space (N finite)
  and orthogonal basis

• Must marginalize to single-time probabilities of
  quantum mechanics diagonal entries of ? and U?U-1

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Axiom Symmetry
D is invariant under relabeling of basis states
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Axiom Indifference
If U acts on and is the identity on H2, then S
should also be the identity on H2 Can formalize
without tensor products partition U into minimal
blocks of nonzero entries
Not the same as commutativity
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Theorem No dynamical theory satisfies both
indifference and commutativity Proof Suppose A
and B share an EPR pair UA applies ?/8
rotation to first qubit, UB applies -?/8 to
second qubit. Consider probability p of being at
00? initially and 10? at the end
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Axiom Robustness

• Small (1/poly(N)) change to ? or U
• ? Small (1/poly(N)) change to joint probabilities
  matrix, Sdiag(?)
• Arguably thats needed for any physical theory or
  model of computation

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Example 1 Product Dynamics
Take probabilities at any two times to be
independent of each other
Symmetric, robust, commutative, but not
indifferent
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Example 2 Dieks Dynamics
Partition U into minimal blocks, then apply
product dynamics separately to each
Symmetric, indifferent, but not commutative or
robust
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Theorem Suppose Then there is a weight-1
flow through the network where flow
through an edge cant exceed the edges capacity
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Proof Idea By the Max-Flow-Min-Cut Theorem
(Ford-Fulkerson 1956), it suffices to show that
any set of edges separating s from t (a cut) has
total capacity at least 1. Let A,B be right,
left edges respectively not in cut C. Then the
capacity of C is so we need to show Fix U
and consider maximum of right-hand side. Equals
the max eigenvalue of a positive semidefinite
matrix, which we can analyze using some linear
algebra
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Example 3 Flow Dynamics
Using the previous theorem, we construct a
dynamical theory that satisfies the symmetry,
indifference, and robustness axioms Not obvious a
priori that any such theory exists
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Model of Computation

• Polynomial-time classical computation, with one
  query to a history oracle

• Oracle takes as input descriptions of quantum
  circuits U1,,UT

• Any dynamical theory D induces a distribution ?D
  over classical histories for

• Oracle chooses a symmetric robust indifferent
  theory D adversarially, then returns a sample
  from ?D

• At least as powerful as standard quantum
  computing

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The Graph Isomorphism Problem

• Decide whether two graphs G and H are isomorphic
• The best known algorithm takes about
  time n number of vertices
• But we dont think Graph Isomorphism is
  NP-complete
• Intuitively, its only as hard as counting
  collisions in
• Could be easier than finding a needle in a
  haystack!

?
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The Collision Problem
3 6 1 5 4 2 vs. 6 2 2 5 6 5

• Given a list of N numbers x1,,xN, youre
  promised that either every number occurs once, or
  every number occurs twice. Decide which.
• Best classical algorithm makes queries
  (birthday paradox)
• Brassard, Høyer, Tapp 1997 gave a quantum
  algorithm that makes N1/3 queries
• Is there a faster quantum algorithmsay, log N
  queries? If so, wed get a polynomial-time
  quantum algorithm for Graph Isomorphism!

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The Collision Problem (cont)

• Aaronson 2002 Any quantum algorithm needs at
  least N1/5 queries
• Improved by Shi to N1/3 queries
• Previously, couldnt even rule out constant
  number of queries!
• Proofs use multivariate polynomials
• Implications
• No dumb quantum algorithm for Graph
  Isomorphism
• Oracle separation between the complexity
  classes BQP (Bounded-Error Quantum
  Polynomial-Time) and DQP (Dynamical Quantum
  Polynomial-Time)

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NP
ConjecturedWorld Map
Satisfiability, Traveling Salesman, etc.
DQPMy New Class
Graph Isomorphism
Approximate Shortest Vector
Factoring
BQPQuantumPolynomialTime
PPolynomial Time
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Solving the Collision Problem by Sampling
Histories
Suppose every number occurs twice. Then
Measurement of 2nd register
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Solving the Collision Problem by Sampling
Histories (cont)
Theorem Under any dynamical theory satisfying
the symmetry and indifference axioms, the first
Fourier transform makes the hidden variable
forget whether it was at i? or j?. So after
the second Fourier transform, it goes to i? half
the time and j? half the time thus with ½
probability we see both i? and j? in the
history Proof Idea Use symmetry axiom, together
with automorphisms of Indifference axiom needed
to trace out second register
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Finding a Marked Item in N1/3 Queries
Probability of observing the marked item after T
iterations is T2/N
N1/3 iterations of Grovers quantum search
algorithm
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N1/3 Search Algorithm Is Optimal

• Bennett, Bernstein, Brassard, Vazirani 1996 If
  a quantum computer searches a list of N items for
  a single randomly-placed marked item, the
  probability of observing the marked item after T
  steps is at most
• So probability of observing it in a history of
  the first T steps is at most

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• Summary If your whole life flashed before you
  in an instant, and if youd prepared for this by
  putting your brain in certain superpositions,
  then (under reasonable axioms) you could solve
  Graph Isomorphism in polynomial time

• But probably still not Satisfiability

• Contrast Nonlinear quantum mechanics would put
  Satisfiability and even harder problems in
  polynomial time (Abrams and Lloyd 1998)

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• Postulate NP-complete problems cant be
  efficiently solved in physical reality

• Justification for the postulate Maybe Im
  wrong, but then Id be too busy solving
  NP-complete problems to care that I was wrong

(1) Quantum states evolve linearly
(2) We cant make unlimited-precision measurements
(3) The self-sampling anthropic principle
(Bostrom 2000) is false
(4) Constraints on quantum gravity?

• The postulate does not imply your whole life
  couldnt flash before you in an instant